Three-Dimensional Unsteady Flow With Heat and

K. N. Lakshmisha S. Venkateswaran1 Department of Aerospace Engineering, Indian institute of Science, Bangalore 560 012, India

G. Nath Department of Applied Mathematics, Indian Institute of Science, Bangalore 560 012, India

1

Three-Dimensional Unsteady Flow With Heat and iass Transfer Over a Continuous Stretching Surface A numerical solution of the unsteady boundary layer equations under similarity assumptions is obtained. The solution represents the three-dimensional unsteady fluid motion caused by the time-dependent stretching of aflat boundary. It has been shown that a self-similar solution exists when either the rate of stretching is decreasing with time or it is constant. Three different numerical techniques are applied and a comparison is made among them as well as with earlier results. Analysis is made for various situations like deceleration in stretching of the boundary, mass transfer at the surface, saddle and nodal point flows, and the effect of a magnetic field. Both the constant temperature and constant heat flux conditions at the wall have been studied.

Introduction

During many mechanical forming processes, such as extrusion, melt-spinning, etc., the extruded material issues through a die. The ambient fluid condition is stagnant but a flow is induced close to the material being extruded, due to the moving surface. In regions away from the slit the flow may be considered to be of boundary layer type, although this is not true just near the slit. Similar situations prevail during the manufacture of plastic and rubber sheets where it is often necessary to blow a gaseous medium through the not-yetsolidified material, and where the stretching force may be varying with time. Another example that belongs to the above class of problems is the cooling of a large metallic plate in a bath, which may be an electrolyte. In this case the fluid flow is induced due to shrinking of the plate. Glass blowing, continuous casting, and spinning of fibres also involve the flow due to a stretching surface. In all these cases, a study of the flow field and heat transfer can be of significant import since the quality of the final product depends to a large extent on the skin friction coefficient and the surface heat transfer rate. After a pioneering work by Sakiadis (1961a; 1961b) the flow field over a continuous stretching surface has drawn considerable attention and a good amount of literature has been generated on this problem (Griffin and Thome, 1967; Vleggar, 1977; Gupta and Gupta, 1977; Abdelhafez, 1986). Most of these papers have presented analytic solutions by which only simple cases are treated. Recently, Wang (1984) has presented an exact similarity solution for the steady threedimensional flow over the stretching flat surface. However, he has not considered the heat transfer aspect. Later, Dutta et al. (1985) analyzed the temperature field with a constant heat flux condition at the wall for steady two-dimensional flow using Crane's (1970) boundary layer solution. Practical situations cited above call for a complete analysis of the fluid dynamics that would be a three-dimensional time-dependent flow with heat and mass transfer at the boundary. It appears that an understanding of the effect of magnetic field on the temperature field is useful during the cooling process in the presence of an electrolytic bath, as mentioned earlier. A problem of this kind would involve considerable mathematical difficulties in numerically solving the governing equations, let alone obtaining a closed-form solution, as pointed out by 1 Currently at the Institute of Hydraulic Research, The University of Iowa, Iowa City, IA 52242. Contributed by the Heat Transfer Division for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received by the Heat Transfer Division February 10, 1987. Keywords: Mass Transfer, Moving Boundaries, Transient and Unsteady Heat Transfer.

Wang (1984). The present paper aims at a comprehensive study of the problem, taking into consideration the various complexities of the process. Three different numerical techniques have been applied and a comparison is made among them and also with the results of Wang. The solutions show excellent agreement with each other and thus confidence in the results is established. Transient behavior is studied in the form of time-dependent stretching of the boundary. It has been shown by asymptotic analysis that a similarity solution is admissible when either the rate of stretching is decreasing or it is constant with respect to time. Thus, four independent variables x, y, z, t are collapsed into a single independent similarity variable 77, and a two-point boundary value problem is formulated. It is found that the shooting method and quasilinearization are extremely sensitive to the initial guesses in cases of unlike stretching (c0) they are the stretching counterparts of the results of Teipel (1979). 2

Description of the Problem

We consider the laminar motion of a viscous, incompressible, electrically conducting fluid caused by the stretching of an infinite flat surface in two lateral directions x and y (see Fig. 1). The surface is assumed to be highly elastic and porous and is stretched by the action of uniform but increasing forces in the same or in opposite directions. The rate of stretching a and b in the two directions varies inversely as a linear function of time. The fluid is assumed to have constant properties. The fluid is at rest at infinity and the no-slip condition is imposed at the stretching surface, where suction or injection can be applied. The temperature of the quiescent fluid is kept constant. In the constant wall temperature (CWT) case the surface

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r,=z(a/v)i/2(l-\t*)~]/1, u= v=

\t*

Pr = v/a

/ ' " + (f+s)f" ~{f')2s'" + (f+s)s"

Mf> - X (/' + / " V/2) = 0

-(s'f-Ms'

g" +(f+s-n\/2)g'T>r

-\(s'

+s'V2) = 0

(10) (11) (12a)

= 0 forCWT for CHF

(126)

The conditions at the surface (-q = 0) and in the quiescent interior (rj—oo) reduce to *? = 0: f=fw,s g=l

(2)

The associated initial and boundary conditions are given by u(x,y,z,0)

(l-Ar)1/2£('/)

to equations (l)-(8) and find that equation (1) is satisifed identically. Equations (2)-(4) reduce on substitution to

= 0, / ' =1, s' =c forCWT

g'=-\

(3)

energy: T, + uTx + vTy + wTz = a.Tzz

1/2

g " + (f+ s - rfK/2)g' Pr + XgPr/2 = 0

With the usual boundary layer approximations the governing equations for the situation described in the previous section are:

ymomentum:

forCWT

,

b h

at

,

rj-oo;

for CHF

f'=s'=g

(13)

=0

(14)

The factor fw represents the mass transfer at the surface with the following relationship:

fw =

-ww{\-\t*y/2/(avy

(15)

(/*„ < 0 for blowing fw will be a constant if wwoc(l -\t*)~ and/,„ > 0 for suction at the surface). The stretching ratio c can have values between - 1 and + 1. When c= 1, the problem is axisymmetric and when c = 0, we have a two-dimensional case, c takes negative values when the natures of the forces in the x and y axes are opposite to each other. In this situation (c 0 it is a nodal point flow. For c< - 1, the equations are insoluble (Davey, 1961) while for c > + 1, we have to simply interchange the x and y axes.

t = time t* = nondimensional time T = temperature V, W = velocity components in x, y, and z directions x,y = rectangular coordinates parallel to the surface z = coordinate perpendicular to the surface a = thermal diffusivity 0 = constant V = independent similarity coordinate a = electrical conductivity ^ = dynamic viscosity P = fluid density V = kinematic viscosity X = parameter associated with unsteadiness

nondimensional temperature shear stresses on x—y plane in x and y directions, respectively Subscripts / = initial conditions at t=0 w = wall condition x, y, z, t = differentiation with respect to x, y, z, and / oo = conditions at the interior of the fluid Superscripts ' = denotes differentiation with respect to TJ AUGUST 1988, Vol. 110/591

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For t* = 0 (X = 0) equations (10)-(12) reduce to those of steady flow and for ^*>0 (X^O) it applies to unsteady flow. Therefore, X is a measure of unsteadiness, and X< 0 means the stretching rate decreases with time while for X>0 it increases with time. Expression for wall shear stresses: T»U=O)= -vax(\~\t*)-^{a/vynf"{0)

(16)

rzyU=0) = -fiayil-Xn-^a/py^s-iO)

(17)

Expression for heat transfer coefficient: h= -k(a/v)W2(\-Xt*)~W2g'(0)

forCWT

(18)

Evidently, f"(0) and s"(0) are measures of wall shear stresses in the x and y directions, respectively. g'(0) represents the heat transfer coefficient for CWT and g(0) is wall temperature for CHF. 3.1 Asymptotic Solution. In order to ascertain the admissible range of values for X we consider the asymptotic behavior of equations (10)-(12), i.e., the behavior of the equations for large »j. Consequently, we set f=C1+F,

s = C2 + S,

g = G, /3 = C , + C 2

(19)

where C, and C 2 are constants and F, S, and G are small. Also from the boundary conditions (14), we get /-C,,

s-C2, F-0,

S-0,

G-Oasij-oo

(20)

Substituting for / , s, and g in equations (10)-(12) and linearizing we get F'" + (/3 - \r)/2)F" - (M+ X)F' = 0

(21)

5"" + ( ( 3 - X ) ? / 2 ) S " - ( M + X ) S ' = 0

(22)

G"+Pr(/3-X»)/2)G'=0 for CWT

(23)

G " + Pr(j3 - XV2)G' + (XPr/2)G = 0

for CHF

(24)

Equations (21) and (22) are identical and equation (24) is also similar to (21) or (22). Hence we consider equations (21) and (23). Applying the transformation F'=exp[-((37)-Xr) 2 /4)/2]i/

(25)

to equation (21), we get H" - [(3X/4 + M) + ((3-Xrj/2) 2 /4]//=0

(26a)

#(°o)-0

(26b)

Equation (26a) is a Weber-type equation whose solution for large -q satisfying the boundary condition (26b) can be expressed in terms of parabolic cylinder functions as (Whittaker and Watson, 1965) W=exp[-(/3-Xij/2) 2 /4](^-XV2)-( 3 / 4 ? ' + M + 1 / 2 'P 1 (v)

(27)

From equations (25) and (27) we get F' = e x p ( - [/32 + /3(2- X)T; - X ( 2 - X)r, 2 /4]/4) ((3-Xr)/2)-( 3 / 4 X + M + 1 / 2 )p,(^)

(28)

where P 1 (i ? ) = l - 2 - ' ( 3 / 4 X + M+l/2)(3/4X + M + 3 / 2 ) (,3-XT)/2)- 2 + 0(/3-Xr;/2)- 4 + . . .

(29)

The asymptotic solution of equation (22) is also given by equation (28). The asymptotic solution of equation (24) is similar to that of (21) or (22) and can be represented by G = exp( - [(32Pr2 + /3Pr(2 - XPr)»j - XPr(2 - XPr)r)2/4]/4]) • [Pr(i3 - Xr?/2)] 0 and the results agreed very well with those obtained by other methods (see Table 1). A step size of Ac = 0.001 was required to obtain the initial slopes accurate up to the first four decimal places. Thus confidence in the results for c < 0 was established. It may be remarked here that the difficulties experienced with negative values of c are not surprising. Davey (1961) has shown that, in the case of a three-dimensional incompressible 10 flow at a stagnation point, the boundary layer equations admit nonunique solutions when - 1 < c < 0 . He has also found that Fig. 2 D i m e n s i o n l e s s w a l l t e m p e r a t u r e v a r i a t i o n w i t h Pr part of the boundary layer begins to show a reversal of flow at (c = / „ = X = M = 0) c = — 0.4294 for that problem. Indeed, as c is increased further in the negative direction, the tendency for the flow reversal increases due to the highly adverse pressure gradient, and there is an increased inflow in the y direction. Further research in 5 Results and Discussion this direction by Libby (1967) has asserted the nonunique A comparison is made among the three different methods as nature of the solutions for — K c < 0 in the case of a comwell as with the results of Wang (1984) (see Table 1) and of pressible boundary layer. In the current problem, perhaps the Dutta et al. (1985) (see Fig 2). The wall shear stresses, heat solutions for - l < c < - 0 . 2 5 latch onto a singular solution transfer coefficient, and wall temperature show very close when the shooting method and quasilinearization are applied. agreement. Consequently, the effect of various parameters on This probably is the reason that the solutions for c< - 0 . 2 5 do the solutions was studied. not converge when either the shooting method or 5.1 Effect of Stretching Ratio c. Heretofore only the quasilinearization is used. Since the method of parametric difpositive values (0 to 1) have been considered by the previous ferentiation is essentially a continuation process, the solutions workers, and in this range solutions were obtained without are marched in the direction of a particular branch and a conany problem by both the shooting method and quasilineariza- verged solution can be obtained. tion. However, in many instances the two lateral forces acting Figure 3 shows some representative velocity and on the sheet are of opposite kind, as for example when a temperature profiles for different stretching ratios. Also, it polymer sheet issues from a die there is contraction in lateral can be seen from Figs. 4 and 5 that increasing c on the positive 1

1

- Present

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Table 2 Comparison of the techniques for unsteady flow (c = fw = 0.5, Pr = 0.7, Ai, = 0.01) Quasilinearization

Parametric differentiation (AX - 0.025)

0.00

-0.25

-0.50

-0.75

-1.00

M =0

M = 1.0

f"(0)

-1.3791017

-1.7529570

s"(0)

-0.6175051

-0.8224046

g'(0)

-0.8039143

-0.7535454

M =0

f"(0)

-1.3119016

-1.6973704

-1.3119180

-1.6973995

s"(0)

-0.5808346

-0.7929573

-0.5808372

-0.7929761

g'(0)

-0.8702118

-0.8274119

-0.8708332

-0.8282535

f"(0)

-1.2445297

-1.6411696

-1.2445405

-1.6412273

s"(0)

-0.5440580

-0.7631492

-0.5440649

-0.7631876

g'(0)

-0.9262852

-0.8878101

-0.9271618

-0.8889399

f"(0)

-1.1769641

-1.5844207

-1.1769913

-1.5845049

s"(0)

-0.5071910

-0.7330250

-0.5072011

-0.7330760

g'(0)

-0.9761054

-0.9406090

-0.9771165

-0.9418685

f"(0)

-1.1092514

-1.5271802

-1.1092886

-1.5272886

s"(0)

-0.4702408

-0.7026060

-0.4702548

-0.7026711

g'(0)

-1.0215186

-0.9882737

-1.0226082

-0.9895902

Fig. 5 Effect of magnetic force on initial slopes i ratios {tw = 0, X = - 0.5, Pr = 0.7)

-0.8

-